s C(K)) (C(K1))* La(C(K1), C(K)) X=(C(K1))*=M(K1), B(X) fEextB(C(K,X)) f(k)EextB(X) kEK. B(X) C(K, X) C(K, .LP(H)) Extreme operator-valued continuous maps

Extreme operator-valued continuous maps

R. Grz~glewicz

Abstract. Let .oq~(H) denote the space of operators on a Hilbert space H. We show that the extreme points of the unit ball of the space of continuous functions

C(K, .LP(H))

(K-compact Hausdorff) are precisely the functions with extremal values. We show also that these extreme points are (a) strongly exposed if and only if dim H < ~ and card K<,~, (b) exposed if and only if / / is separable and K carries a strictly positive measure.

1. Introduction

Let

C(K, X)

denote the Banach space o f all continuous functions from a c o m p a c t Hausdorff space K to a Banach space X equipped with the s u p r e m u m n o r m Ilfll=supk~x llf(k)]l. By

B(X)

we denote the unit ball of the Banach space X.

There is a natural conjecture a b o u t extreme points:

( . )

fEextB(C(K,X))

if and only if

f(k)EextB(X)

for all

kEK.

Obviously the " i f p a r t " in ( . ) always holds. This conjecture has been proved to be true under various additional assumptions. We list some o f the known results.

T h e conjecture is true if:

1) X is strictly convex, 2)

B(X)

is polytope,

3)

X=(C(K1))*=M(K1),

where K1 is a c o m p a c t H a u s d o r f f space ([31], T h e o r e m 4, 5). N o t e that in this case the conjecture ( . ) is equivalent to the fact that extreme operators in the unit ball o f the space o f c o m p a c t operators

La(C(K1), C(K))

are nice in the sense o f Morris and Phelps. F o r other similar results on the space o f weak* continuous functions from K into

(C(K1))*

(this corresponds to the case o f extreme point of the unit ball o f the space o f all bounded linear operators

s C(K))

(see [6], p. 490)) under various assumptions on K and Kx (see [2], [1], [34], [9], [20], see also [35], [36] for negative examples).

4) X has 3.2.I.P. ([37]).

74 R. Grz~lewiez

5)

B(X)

is stable (to get it we use (iii')) in [4] and Michael's selection Theorem [29]).

6) X=LX(#) ([39]).

7) X=L~'(It) is an Orlicz space equipped with the Luxemburg norm ([18]).

8)

X=M(Ka,Z)

is the space of Z-valued regular Borel measures of finite varia- tion, Z is a Banach space,/(1 is a compact Hausdorff space ([40]).

The conjecture ( . ) is not true for every Banach space X. A negative example was given by Blumental, Lindenstrauss and Phelps for

X=R 4

equipped with a suitable norm ([2]). The class of finite dimensional spaces for which conjecture ( . ) holds was described by Papadopoulou ([32]). She proved that

B(X)

(dim X<oo) is stable if and only if all k-skeletons (k = 0, 1 ... n) of B (X) are closed (a k-skeleton of

B(X)

is a set of all

xEB(X)

such that the face generated by x has dimension less than or equal to k).

After dealing with the conjecture for Banach spaces the next natural step is to consider X as a space of linear operators. We denote by .LP(Y) the space of all bounded linear operators from a Banach space Y into itself equipped with the operator norm.

For X=Lz~ l < p < ~ o , p ~ 2 , the conjecture does not holds. Indeed, we have the following example. Let (a,)El v be such that II(an)Jlp=l and a , > 0 for all n. We define fEC([0, 1], .~~ by

f(k)=Tk,

kE[0, 1] where

Tk((x.)) = x , ( ~ , ~/af + ag, ~c-~. [/o7 + ag, a a, a, .... ),

(x,)El p. We have for p > 2 ,

TkEextB(Le(IV))

for kE(0, 1) and TkqextB(~(/P)) for k = 0 or 1 (see [15], see also [14]). Using adjoint operators an analogous ex- ample can be written for pE(1, 2).

In this note we consider the case p = 2 . It turns out that the conjecture ( . ) holds for the space of operators acting on a Hilbert space (Section 2).

In Section 3 we consider under what conditions elements of ext

B(C(K,

5e(H))) are exposed or strongly exposed. We show that extreme points of

B(C(K,

.oq'(H))) are (a) strongly exposed if and only if dim H < ~ and card K < ~o,

(b) exposed if and only if H is separable and K carries a strictly positive measure.

2. Extreme points in

B(C(K,

(H)))

Let H be a (real or complex) Hilbert space. We denote by PrE.oq'(H) the orthogonal projection onto a subspace E of H. The set of extreme points ext B (.oq' (H)) coincides with the set of all isometries and coisometries (see [23] for the complex case, and [16] for the real case). Our aim is to characterize ext

B(K, ~(H)).

Note that the below presented Theorem was proved in [14, p. 314, ( . . ) ] in the case when H is finite dimensional.

Extreme operator-valued continuous maps 75 In the finite dimensional case the dual of LZ(H) was also considered, and it turns out that the unit ball of s is stable (see [17]), so the conjecture ( . ) also holds for C(K, B(LZ(H)*)) (dim H < ~).

Now we recall some facts we will use in the proof of Theorem 1 below.

Let TC s (H) be a contraction. We put

M ( T ) = {hEn: [IZhll = Ilhll},

We have Z*Th=h for h~M(T). Moreover T ( M ( T ) ) = M ( T * ) and T ( M • M•

Obviously, if T~extB(.,q'(H)) then neither T T * = I nor T * T = I i.e.

M • (T) # {0} # M • (T*).

Put S = ( ( I + T*T)/2) x/2. We have I/2<=S<=I. Therefore S -1, S 1/2 and S -x/~

exist. We have

0~$1/2<-L

Hence

0 <= S~/2(2Sa/Z-I) = 2 S - S ~/2 <- L Since T*T<=S 2 we have

llTxII ~ = (T*Zx, x) <= (S2x, x) = !15x112.

So [ITS-1I[<_-I. We get

IITS-X/ell ~_ [ITS-lit [lSa/2ll <_- 1 and

[IT(2I-S-a/2)[I <-[IZS-l[I 1[2S-St/~l[ ~ 1.

Theorem 1. Let H be a Hilbert space and let K be a compact Hausdorff space. Then f ~ ext B(C(K, s (H)))

i f and only i f

f(k)~extB(LZ(H)) for all kCK.

Proof. Let f~B(C(K, .La(H))). Assume that f(ko) fails to be extremal for some koCK, i.e.

M • (f(ko)) # {0} # M • (f(ko)).

We need to prove that then

f r ext B(C(K, ~ (H))).

Put T k=f(k). We consider two cases.

_ - 1 l ~

1 ~ There exist ko~K and xCMX(Tko) such that Tk0X~0. Put A(k)--TkSk , and f2(k)=2Tk-TkS[XS~/2, where Sk=((I+Tk*Tk)/2) u2. Obviously

ae(H))) and f = (A+A)/2.

Since

M(Tko ) = M(T~oTko ) = M(S~o ) = M(S~o) = M(S~k/o ~)

76 R . G r z ~ l e w i e z

and

Tk, T~xEM (S~,)

* .t 11~ we have

SV2T,* T,

ko ko ko-~, <

IIZ~Tkoxll.

Since

S~ 12

and Tk*T k commute, we get

9 /2 *

TioTkoS~ko x ~ T~oTkox,

so

TkoS~k~x ~ Tko x.

Hence

Tko~TkoS~l/*

and

fl(ko)~f2(ko)

i.e. f is not extreme.

2 ~

Tk(M•

= {0} =

T~(M•

for all

k~K.

Then Tk*Tk=P~(T~ ). Therefore

k-~PM(r, )

is a continuous function, so

k-~PM•

is a continuous function, too. Fix

koEK

such that

f(ko)r

Choose

eEM • (f(ko))

and

gEM •

(f(k0)*) with [[ei[ =[]gl] = 1. We have

IITk + PM•174 <= 1.

Hence f is not extreme.

3. Exposed and strongly exposed points in

B(C(K, (H)))

A point q0 in a convex set Q of a (real or complex) Banach space E is said to be exposed if there exists a bounded R-linear functional ~: E-~R such that ~ (q0) >

~(q) for all

qEQ\{qo}.

An exposed point

qoEQ

is called strongly exposed if for any sequence q.E Q the condition

~(q.)~(qo)

implies llq.-q0]l-~0. Obviously each exposed point is extreme.

Note that extreme points of

B(.~(H))

are strongly exposed in and only if H is a finite dimensional Hilbert space, and exposed but not strongly exposed if and only if H is separable infinite dimensional. Moreover there are no exposed points in

B(.~(H))

if and only if H is not separable ([19]).

Consider the Bochner LP-space ( l < p < oo). For

f~LP(ll, X) (X

is a Banach space), if I l f [ ] = l and

f(t)/llf(t)llEextB(X)

for t~suppf/~-a.e., then

f C e x t

B(LP(I~, X)).

Generally the converse does not holds. A negative example was given by Greim [12]

for a nonseparable Banach space X. But for all separable Banach spaces X this property characterizes extreme functions (see [38], [21], [22], [13]). For the strongly exposed points the analogous natural condition on the values o f f are sufficient, whenever X is smooth ([10], see also [11]). Recently W. Kurtz considered strongly exposed points in Bochner--Orlicz spaces [25] (see also [26]). A compact Hausdorff space K is said to carry a strictly positive measure, if there exists a strictly positive Radon measure # on X (i.e. p ( q / ) > 0 for all non-empty open subsets q/ of X).

Extreme operator-valued continuous maps 77 Several authors have workes on the problem of the characterization of spaces X which carry a strictly positive measure. Maharam [28] has given necessary and sufficient conditions for the existence of strictly positive measures. Kelley considers strictly positive measures on Boolean algebras. In his work [24] he introduced the notion o f the intersection number of a collection of subsets to give a characteriza- tion of spaces which carry a strictly positive measure.

Next it turns out that the countable chain condition is not sufficient for the existence of such a measure (see Gaifman [8]). Note that in the case of a compact Hausdorff space, the problem mentioned above is equivalent to the problem of existence of a finitely additive strictly positive measure. Rosenthal ([33], Th. 4.5b) proved that C (X) carries a strictly positive functional if and only if its dual C (X)*

contains a weakly compact total subset. Other results can be found in [7], [30], [3].

We refer the reader to [5, Chapter 6] for a survey of known results about strictly positive measures. In fact, we can consider a strictly positive measure on X as a functional on

C(X)

which exposes the function 1.

Theorem 2.

I f a Hilbert space H is separable and a Hausdorff compact space K carries a strictly positive measure, then each extreme point of B(C(K,

L~'(H)))

is exposed.

I f H is not separable or K does not carry a strictly positi~'e measure then B(C(K, ~

(H)))

contains no exposed points.

I f H is infinite dimensional then B(C(K,

L~~

contains no strongly exposed points.

Proof.

Suppose that H is separable and K carries a strictly positive measure.

Assume that /z(K)= 1. We fix an orthonormal basis

{ei}iet

in H. Fix foEext

B(C(K,

A~

Put K I = { k E K :

f(k)

is an isometry} and

K 2 = K \ K I .

The set Kx is closed. Let

(ai)tel

be a sequence o f strictly positive reals such that ~ i e t as= 1. We define a functional ~ on

C(K, .~(H))

by

( f ) = f r~ Z , c I a,

Re (If(k)] (e,), [fo (k)] (e,))

d/z

+ f K% Z , e , a,

R e ([f(k)]*(e,),

[fo(k)l*(e,)) d#,

fEC(K, .s

The functional exposesfo in

B(C(K,

.oq~ Indeed ~(f)_-__l=

~(fo) for all

fEB(C(K,

Le(n))).

Suppose that ~ ( f x ) = 1 for some

fxEB(C(K,

La(H))). Because Re ([ fx (k)] (e,), [ fo (k)]

(el)) <= I,

the condition ~ ( f x ) = l implies that

[f~(k)](e,)=[fo(k)](e,)

for all

iEI

and

kEK~

78 R. Gr-z~lewiez

/z-a.e. Analogously f x = f o p-a.e, on Kz. Hence by continuity

fx=fo.

So we finish the proof of the first part of theorem.

Now suppose that a functional ~0 exposes

foEextB(C(K,L#(H))) in B(C(K,

.La(H))). We define a functional m on

C(K)

by

re(h) = ~o(hfo), hEC(K).

We claim that m is strictly positive. Indeed, suppose to get a contradiction, that there exists

hoEC(K )

such that 0 ~ h o ( k ) ~ l , ho~0, and m(h0)<0.

Then

~o((1-ho)fo) = m(1--ho) -<-- re(l) = r

and hofo ~ 0 , which is impossible. It follows that K carries a strictly positive measure.

Consider now a function n on all subsets of 1 defined by

n(L) = ~o(foPl~ te,: ~ L))

L C I

(in the case when all

fo(k)

are coisornetries we define n by n ( L ) =

~0(P~(~,:~cz~f0)). The function n is finitely additive on the family of all subsets o f / . Moreover n ( I ) = l and

n(L)~O

for

L c I .

Suppose that n(L0)--0 for some non-empty L 0 c I . Then

SO

~0(f0~Tte,:~L0 }) = 0,

(i.e. 40 does not expose fo). This contradiction proves that n(L0)>0. Hence I is countable and H is separable, which yields the second part of the theorem.

Let (Li)j~ N be a sequence of non-empty disjoint subsets o f L Then

n ( L j ) ~ O.

Hence

~0(f0e~e ~CLg) = ~o(fo)--n(Lj) 1, ~offo)

and

Ilf0-foe~t,,:~r =

IlfoP~ te,:ie Lsll I = 1.

Therefore fo is not strongly exposed in the case when I is infinite (i.e. H is infinite dimensional). This proves the third part of the theorem.

Theorem 3.

Let H be a finite dimensional Hilbert space and K be a Hausdorff compact space. Then each extreme point of B(C(K,

.L#(H)))

is strongly exposed, if and only if K is finite.

Proof.

Let d i m H < ~ o and c a r d K < r Fix

foEB(C(K,

.La(H))). Let r/k be a functional on .L#(H) which strongly exposesfo(k),

kEK.

It is easy to see that a

Extreme operator-valued continuous maps 79

functional ~ defined by

r = Z k r f ~ C ( K , .C#(H)), exposes ./~.

N o w suppose that card K = ~o. Let a fun ction al 3o expose f0 C ext B (C (K, .C>e (H))).

Let (k.).(N be a sequence o f distinct points o f K such that lira k . = k o . Let h . E C ( K ) , nEN, be such that 0-<__h. <- I, h . ( k . ) = 1 and supp h. c~supp h . = 0 if n1#n2. Put a#=r N o t e that a.=>0 because

~ 0 ( f 0 ) - a. = 3 0 ( ( 2 - h . ) f 0 ) ~ ~0(f0).

F o r every finite subset L o f N we have

Z . e z an = r h.)fo) ~- r

Hence a.---O. Therefore

~ o ( ( 1 - h , ) f 0 ) = ~o(fo)-a~ -~ ~o(fo) and

1I(2 -h.)f0-f011 = l i h . f 11 = 2.

Thus fo is not strongly exposed by 40. This ends the proof.

Remark. I would like to express my thanks to the referee for his useful remarks which shorten the p r o o f o f T h e o r e m 2.

Acknowledgement. Written partially while the author was a research fellow of the Alexander von Humboldt-Stiftung at the Mathematisches Institut der Eberhard- Karls-Universitiit in Tiibingen.

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Received September 26, 1986

Received in revised form November 9. 1989

R. Gr7~tlewicz

Institute of Mathematics Technical University Wyb. Wyspiafiskiego 27

50-370 Wroclaw Poland